1 Programming Principles 1
1.1 Introduction 2
1.2 The Game of Life 4
1.2.1 Rules for the Game of Life 4
1.2.2 Examples 5
1.2.3 The Solution: Classes, Objects, and Methods 7
1.2.4 Life: The Main Program 8
1.3 Programming Style 10
1.3.1 Names 10
1.3.2 Documentation and Format 13
1.3.3 Refinement and Modularity 15
1.4 Coding, Testing, and Further Refinement 20
1.4.1 Stubs 20
1.4.2 Definition of the Class Life 22
1.4.3 Counting Neighbors 23
1.4.4 Updating the Grid 24
1.4.5 Input and Output 25
1.4.6 Drivers 27
1.4.7 Program Tracing 28
1.4.8 Principles of Program Testing 29
1.5 Program Maintenance 34
1.5.1 Program Evaluation 34
1.5.2 Review of the Life Program 35
1.5.3 Program Revision and Redevelopment 38
1.6 Conclusions and Preview 39
1.6.1 Software Engineering 39
1.6.2 Problem Analysis 40
1.6.3 Requirements Specification 41
1.6.4 Coding 41
2 Introduction to Stacks 49
2.1 Stack Specifications 50
2.1.1 Lists and Arrays 50
2.1.2 Stacks 50
2.1.3 First Example: Reversing a List 51
2.1.4 Information Hiding 54
2.1.5 The Standard Template Library 55
2.2 Implementation of Stacks 57
2.2.1 Specification of Methods for Stacks 57
2.2.2 The Class Specification 60
2.2.3 Pushing, Popping, and Other Methods 61
2.2.4 Encapsulation 63
2.3 Application: A Desk Calculator 66
2.4 Application: Bracket Matching 69
2.5 Abstract Data Types and Their Implementations 71
2.5.1 Introduction 71
2.5.2 General Definitions 73
2.5.3 Refinement of Data Specification 74
3 Queues 78
3.1 Definitions 79
3.1.1 Queue Operations 79
3.1.2 Extended Queue Operations 81
3.2 Implementations of Queues 84
3.3 Circular Implementation of Queues in C++ 89
3.4 Demonstration and Testing 93
3.5 Application of Queues: Simulation 96
3.5.1 Introduction 96
3.5.2 Simulation of an Airport 96
3.5.3 Random Numbers 99
3.5.4 The Runway Class Specification 99
3.5.5 The Plane Class Specification 100
3.5.6 Functions and Methods of the Simulation 101
3.5.7 Sample Results 107
4 Linked Stacks and Queues 112
4.1 Pointers and Linked Structures 113
4.1.1 Introduction and Survey 113
4.1.2 Pointers and Dynamic Memory in C++ 116
4.1.3 The Basics of Linked Structures 122
4.2 Linked Stacks 127
4.3 Linked Stacks with Safeguards 131
4.3.1 The Destructor 131
4.3.2 Overloading the Assignment Operator 132
4.3.3 The Copy Constructor 135
4.3.4 The Modified Linked-Stack Specification 136
4.4 Linked Queues 137
4.4.1 Basic Declarations 137
4.4.2 Extended Linked Queues 139
4.5 Application: Polynomial Arithmetic 141
4.5.1 Purpose of the Project 141
4.5.2 The Main Program 141
4.5.3 The Polynomial Data Structure 144
4.5.4 Reading and Writing Polynomials 147
4.5.5 Addition of Polynomials 148
4.5.6 Completing the Project 150
4.6 Abstract Data Types and Their Implementations 152
5 Recursion 157
5.1 Introduction to Recursion 158
5.1.1 Stack Frames for Subprograms 158
5.1.2 Tree of Subprogram Calls 159
5.1.3 Factorials: A Recursive Definition 160
5.1.4 Divide and Conquer: The Towers of Hanoi 163
5.2 Principles of Recursion 170
5.2.1 Designing Recursive Algorithms 170
5.2.2 How Recursion Works 171
5.2.3 Tail Recursion 174
5.2.4 When Not to Use Recursion 176
5.2.5 Guidelines and Conclusions 180
5.3 Backtracking: Postponing the Work 183
5.3.1 Solving the Eight-Queens Puzzle 183
5.3.2 Example: Four Queens 184
5.3.3 Backtracking 185
5.3.4 Overall Outline 186
5.3.5 Refinement: The First Data Structure and Its Methods 188
5.3.6 Review and Refinement 191
5.3.7 Analysis of Backtracking 194
5.4 Tree-Structured Programs: Look-Ahead in Games 198
5.4.1 Game Trees 198
5.4.2 The Minimax Method 199
5.4.3 Algorithm Development 201
5.4.4 Refinement 203
5.4.5 Tic-Tac-Toe 204
6 Lists and Strings 212
6.1 List Definition 213
6.1.1 Method Specifications 214
6.2 Implementation of Lists 217
6.2.1 Class Templates 218
6.2.2 Contiguous Implementation 219
6.2.3 Simply Linked Implementation 221
6.2.4 Variation: Keeping the Current Position 225
6.2.5 Doubly Linked Lists 227
6.2.6 Comparison of Implementations 230
6.3 Strings 233
6.3.1 Strings in C++ 233
6.3.2 Implementation of Strings 234
6.3.3 Further String Operations 238
6.4 Application: A Text Editor 242
6.4.1 Specifications 242
6.4.2 Implementation 243
6.5 Linked Lists in Arrays 251
6.6 Application: Generating Permutations 260
7 Searching 268
7.1 Searching:Introduction and Notation 269
7.2 Sequential Search 271
7.3 Binary Search 278
7.3.1 Ordered Lists 278
7.3.2 Algorithm Development 280
7.3.3 The Forgetful Version 281
7.3.4 Recognizing Equality 284
7.4 Comparison Trees 286
7.4.1 Analysis for n = 10 287
7.4.2 Generalization 290
7.4.3 Comparison of Methods 294
7.4.4 A General Relationship 296
7.5 Lower Bounds 297
7.6 Asymptotics 302
7.6.1 Introduction 302
7.6.2 Orders of Magnitude 304
7.6.3 The Big-O and Related Notations 310
7.6.4 Keeping the Dominant Term 311
8 Sorting 317
8.1 Introduction and Notation 318
8.1.1 Sortable Lists 319
8.2 Insertion Sort 320
8.2.1 Ordered Insertion 320
8.2.2 Sorting by Insertion 321
8.2.3 Linked Version 323
8.2.4 Analysis 325
8.3 Selection Sort 329
8.3.1 The Algorithm 329
8.3.2 Contiguous Implementation 330
8.3.3 Analysis 331
8.3.4 Comparisons 332
8.4 Shell Sort 333
8.5 Lower Bounds 336
8.6 Divide-and-Conquer Sorting 339
8.6.1 The Main Ideas 339
8.6.2 An Example 340
8.7 Mergesort for Linked Lists 344
8.7.1 The Functions 345
8.7.2 Analysis of Mergesort 348
8.8 Quicksort for Contiguous Lists 352
8.8.1 The Main Function 352
8.8.2 Partitioning the List 353
8.8.3 Analysis of Quicksort 356
8.8.4 Average-Case Analysis of Quicksort 358
8.8.5 Comparison with Mergesort 360
8.9 Heaps and Heapsort 363
8.9.1 Two-Way Trees as Lists 363
8.9.2 Development of Heapsort 365
8.9.3 Analysis of Heapsort 368
8.9.4 Priority Queues 369
8.10 Review: Comparison of Methods 372
9 Tables and Information Retrieval 379
9.1 Introduction: Breaking the lg n Barrier 380
9.2 Rectangular Tables 381
9.3 Tables of Various Shapes 383
9.3.1 Triangular Tables 383
9.3.2 Jagged Tables 385
9.3.3 Inverted Tables 386
9.4 Tables: A New Abstract Data Type 388
9.5 Application: Radix Sort 391
9.5.1 The Idea 392
9.5.2 Implementation 393
9.5.3 Analysis 396
9.6 Hashing 397
9.6.1 Sparse Tables 397
9.6.2 Choosing a Hash Function 399
9.6.3 Collision Resolution with Open Addressing 401
9.6.4 Collision Resolution by Chaining 406
9.7 Analysis of Hashing 411
9.8 Conclusions:Comparison of Methods 417
9.9 Application:The Life Game Revisited 418
9.9.1 Choice of Algorithm 418
9.9.2 Specification of Data Structures 419
9.9.3 The Life Class 421
9.9.4 The Life Functions 421
10 Binary Trees 429
10.1 Binary Trees 430
10.1.1 Definitions 430
10.1.2 Traversal of Binary Trees 432
10.1.3 Linked Implementation of Binary Trees 437
10.2 Binary Search Trees 444
10.2.1 Ordered Lists and Implementations 446
10.2.2 Tree Search 447
10.2.3 Insertion into a Binary Search Tree 451
10.2.4 Treesort 453
10.2.5 Removal from a Binary Search Tree 455
10.3 Building a Binary Search Tree 463
10.3.1 Getting Started 464
10.3.2 Declarations and the Main Function 465
10.3.3 Inserting a Node 466
10.3.4 Finishing the Task 467
10.3.5 Evaluation 469
10.3.6 Random Search Trees and Optimality 470
10.4 Height Balance: AVL Trees 473
10.4.1 Definition 473
10.4.2 Insertion of a Node 477
10.4.3 Removal of a Node 484
10.4.4 The Height of an AVL Tree 485
10.5 Splay Trees: A Self-Adjusting Data Structure 490
10.5.1 Introduction 490
10.5.2 Splaying Steps 491
10.5.3 Algorithm Development 495
10.5.4 Amortized Algorithm Analysis:Introduction 505
10.5.5 Amortized Analysis of Splaying 509
11 Multiway Trees 520
11.1 Orchards, Trees, and Binary Trees 521
11.1.1 On the Classification of
Species 521
11.1.2 Ordered Trees 522
11.1.3 Forests and Orchards 524
11.1.4 The Formal Correspondence 526
11.1.5 Rotations 527
11.1.6 Summary 527
11.2 Lexicographic Search Trees: Tries 530
11.2.1 Tries 530
11.2.2 Searching for a Key 530
11.2.3 C++ Algorithm 531
11.2.4 Searching a Trie 532
11.2.5 Insertion into a Trie 533
11.2.6 Deletion from a Trie 533
11.2.7 Assessment of Tries 534
11.3 External Searching: B-Trees 535
11.3.1 Access Time 535
11.3.2 Multiway Search Trees 535
11.3.3 Balanced Multiway Trees 536
11.3.4 Insertion into a B-Tree 537
11.3.5 C++ Algorithms:Searching and Insertion 539
11.3.6 Deletion from a B-Tree 547
11.4 Red-Black Trees 556
11.4.1 Introduction 556
11.4.2 Definition and Analysis 557
11.4.3 Red-Black Tree Specification 559
11.4.4 Insertion 560
11.4.5 Insertion Method Implementation 561
11.4.6 Removal of a Node 565
12 Graphs 569
12.1 Mathematical Background 570
12.1.1 Definitions and Examples 570
12.1.2 Undirected Graphs 571
12.1.3 Directed Graphs 571
12.2 Computer Representation 572
12.2.1 The Set Representation 572
12.2.2 Adjacency Lists 574
12.2.3 Information Fields 575
12.3 Graph Traversal 575
12.3.1 Methods 575
12.3.2 Depth-First Algorithm 577
12.3.3 Breadth-First Algorithm 578
12.4 Topological Sorting 579
12.4.1 The Problem 579
12.4.2 Depth-First Algorithm 580
12.4.3 Breadth-First Algorithm 581
12.5 A Greedy Algorithm:
Shortest Paths 583
12.5.1 The Problem 583
12.5.2 Method 584
12.5.3 Example 585
12.5.4 Implementation 586
12.6 Minimal Spanning Trees 587
12.6.1 The Problem 587
12.6.2 Method 589
12.6.3 Implementation 590
12.6.4 Verification
of Prim’s Algorithm 593
12.7 Graphs as Data Structures 594
13 Case Study: The Polish Notation 598
13.1 The Problem 599
13.1.1 The Quadratic Formula 599
13.2 The Idea 601
13.2.1 Expression Trees 601
13.2.2 Polish Notation 603
13.3 Evaluation of Polish Expressions 604
13.3.1 Evaluation of an Expression in Prefix Form 605
13.3.2 C++ Conventions 606
13.3.3 C++ Function for Prefix Evaluation 607
13.3.4 Evaluation of Postfix Expressions 608
13.3.5 Proof of the Program: Counting Stack Entries 609
13.3.6 Recursive Evaluation of Postfix Expressions 612
13.4 Translation from Infix Form to Polish Form 617
13.5 An Interactive Expression Evaluator 623
13.5.1 Overall Structure 623
13.5.2 Representation of the Data: Class Specifications 625
13.5.3 Tokens 629
13.5.4 The Lexicon 631
13.5.5 Expressions: Token Lists 634
13.5.6 Auxiliary Evaluation Functions 639
13.5.7 Graphing the Expression: The Class Plot 640
13.5.8 A Graphics-Enhanced Plot Class 643
References for Further Study 645
A Mathematical Methods 647
A.1 Sums of Powers of Integers 647
A.2 Logarithms 650
A.2.1 Definition of Logarithms 651
A.2.2 Simple Properties 651
A.2.3 Choice of Base 652
A.2.4 Natural Logarithms 652
A.2.5 Notation 653
A.2.6 Change of Base 654
A.2.7 Logarithmic Graphs 654
A.2.8 Harmonic Numbers 656
A.3 Permutations, Combinations, Factorials 657
A.3.1 Permutations 657
A.3.2 Combinations 657
A.3.3 Factorials 658
A.4 Fibonacci Numbers 659
A.5 Catalan Numbers 661
A.5.1 The Main Result 661
A.5.2 The Proof by One-to-One Correspondences 662
A.5.3 History 664
A.5.4 Numerical Results 665
B Random Numbers 667
B.1 Introduction 667
B.2 Strategy 668
B.3 Program Development 669
C Packages and Utility Functions 674
C.1 Packages and C++ Translation Units 674
C.2 Packages in the Text 676
C.3 The Utility Package 678
C.4 Timing Methods 679
D Programming Precepts, Pointers, and Pitfalls 681
D.1 Choice of Data Structures and Algorithms 681
D.1.1 Stacks 681
D.1.2 Lists 681
D.1.3 Searching Methods 682
D.1.4 Sorting Methods 682
D.1.5 Tables 682
D.1.6 Binary Trees 683
D.1.7 General Trees 684
D.1.8 Graphs 684
D.2 Recursion 685
D.3 Design of Data Structures 686
D.4 Algorithm Design and Analysis 687
D.5 Programming 688
D.6 Programming with Pointer Objects 689
D.7 Debugging and Testing 690
D.8 Maintenance 690
Index 693
